Integrated research of the general hypergeometric systems and nonlinear integrable systems

一般超几何系统与非线性可积系统的综合研究

• 批准号: 11440058
• 负责人: KIMURA Hironobu
• 金额: $ 8万
• 依托单位: Kumamoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440058/
• 关键词: Painleve equation; space of initial conditions; confluence; general hypergeometric function; de Rham cohomology; intersection theory; generalized Airy function; accessory parameter; Gauss Manin系; Grassmann多様体; Airy関数; generating function; 超幾何関

The objective of this research is 1) the study of the general hypergeometric functions and Okubo systems, 2) the study of nonlinear integrable systems including Painleve equations. The conjugacy classes of the centralizers of regular elements of GL(N,C) are determined by partitions of N. The general hypergeometric functions are functions on the Grassmannian manifold Gr(n,N) obtained by the Radon transformation of characters of universal covering groups of centralizers. We explicitly determined the algebraic de Rham cohomology groups associated with the integral representation of the general hypergeometric functions. This problem has been isolved in the case n=2 and in the case n>2 with the partitions (1,…,1), (N). For the case of partitions (q, 1,…,1), we proved the purity of the cohomology group, determined the dimension of the top cohomology and gave an explicit basis for it. This result will be important in constructing the Gauss-Manin system characterizing the general hypergeometri … More c functions. In the case where the partition is (N), we constructed the intersection theory of de Rham cohomology and expressed the intersection numbers in terms of skew Schur polynomials. In this computation, we recognized that an analogue of flat basis plays an important roles which appears in the theory of singularity. For the differential equation of Schlesinger type on P^1 without accessory parameters, we showed that the solutions have integral representations using the corresponding result for Okubo system. This integral representation is a particular case of that of GKZ hypergeometric functions. Thus it may be an interesting problem to understand the accessory parameter free equations in the framework of GKZ hypergeometric functions and to generalize this problem to the equations with irregular singularities.For the Painleve equations, we showed that there is a symplectic structure for the space of initial conditions for each Painleve equation and also showed that the geometry of the space of initial conditions determines the Painleve equation. We found an interesting phenomenon that a generating function for a series of rational solutions of Painleve II coincides with the asymptotic expansion at infinity of the function obtained from Airy function. Less

本论文的主要目的是：1）研究一般超几何函数和大久保系统; 2）研究包括Painleve方程在内的非线性可积系统。GL（N，C）的正则元的中心化子的共轭类由N的分拆决定。广义超几何函数是Grassmannian流形Gr（n，N）上的函数，通过中心化子的泛覆盖群的特征标的Radon变换得到。我们明确地确定了与一般超几何函数的积分表示相关的代数de Rham上同调群。这个问题在n=2和n>2的情况下都得到了解决，其中分区为（1，.，1），（N）。对于（q，1，…，1）的划分，我们证明了上同调群的纯性，确定了上同调的维数，并给出了上同调的一个显式基.这一结果对于构造刻画一般超几何的Gauss-Manin系统具有重要意义 ...更多信息 c函数。在剖分为（N）的情形下，构造了de Rham上同调的交理论，并将交数表示为斜Schur多项式。在这个计算中，我们认识到，一个类似的平坦基起着重要的作用，出现在奇点理论。对于P^1上不含辅助参数的Schlesinger型微分方程，利用Okubo系统的相应结果，证明了其解具有积分表示.这种积分表示是GKZ超几何函数的一种特殊情况。因此，在GKZ超几何函数的框架下理解无附加参数方程，并将其推广到具有非正则奇异性的方程，可能是一个有趣的问题。对于Painleve方程，我们证明了每个Painleve方程的初始条件空间都有一个辛结构，并且初始条件空间的几何形状决定了Painleve方程。我们发现了一个有趣的现象，即Painleve II的一系列有理解的母函数与由Airy函数得到的函数在无穷远处的渐近展开式一致。少

项目成果

K.Takano: "On some Hamiltonian structures of Painleve systems II"J.Math.Soc.Japan. 51. 843-866 (1999)

K.Takano：“关于 Painleve 系统 II 的一些哈密顿结构”J.Math.Soc.Japan。

木村 弘信: "退化Garnier系の初期値空間について"数理解析研究所講究録. 1133. 18-27 (2000)

Hironobu Kimura：“论简并卡尼尔系统的初值空间”数学分析研究所的 Kokyuroku 1133. 18-27 (2000)。

K.Iwasaki, Y.Kamimura: "Inverse bifurcation problem, singular Wiener-Hopf equations and nathematical model in ecology"J. Math. Biol.. 43. 101-143 (2001)

K.Iwasaki，Y.Kamimura：“生态学中的逆分岔问题、奇异 Wiener-Hopf 方程和数学模型”J。

K.Iwasaki: "Intersection matrix of a generalized Airy function in terms of skew Schur polynomials"Proc.Japan Acad., Ser.A Math.Sci.. 76. 135-140 (2000)

K.Iwasaki：“根据斜 Schur 多项式的广义艾里函数的交集矩阵”Proc.Japan Acad.，Ser.A Math.Sci.. 76. 135-140 (2000)

H. Kimura: "On the cohomological intersection, numbers for the generalized Airy integrals"Suurikenkokyuroku. 1296. 63-72 (2002)

H. Kimura：“在上同调交点上，广义艾里积分的数字”Suurikenkokyuroku。